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Computations of higher elliptic units in optimal settings

This paper presents a simplified conjecture for constructing higher elliptic units as special values of higher elliptic Gamma functions over number fields with exactly one complex place, offering a potential solution to Hilbert's 12th problem for such fields and providing computational evidence for degrees 3 through 6.

Original authors: Pierre L. L. Morain

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pierre L. L. Morain

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Map to Hidden Numbers

Imagine you are a cartographer trying to draw a map of a mysterious, invisible kingdom. In the world of mathematics, this kingdom is made of "number fields"—complex extensions of the familiar counting numbers that follow their own strict, hidden rules. For centuries, mathematicians have been obsessed with a specific puzzle known as Hilbert's 12th Problem. It asks a simple but incredibly difficult question: Can we write down a specific recipe to generate all the "symmetrical" (abelian) parts of these number kingdoms?

For a long time, we only had a working recipe for one tiny, special type of kingdom: the imaginary quadratic fields. In these cases, mathematicians use a tool called "elliptic units," which are like magical keys made from a special function called the theta function. These keys unlock the doors to the kingdom's symmetrical extensions. However, for most other number fields—especially those with more complex structures—no one knew how to forge these keys. It was as if we had a map for one small island but were blind to the vast continents surrounding it. This paper steps into that darkness, proposing a new, more powerful set of keys to unlock these harder-to-reach mathematical territories.

The New Keys: Higher Elliptic Units

In this paper, Pierre L. L. Morain suggests a way to construct these missing keys for a specific group of number fields: those that have exactly one "complex place" (a fancy way of saying they have one pair of complex roots and the rest are real). The author proposes that we can build these new keys, called "higher elliptic units," by using a more advanced mathematical tool called the higher elliptic Gamma function.

Think of the old theta function as a simple, single-note flute. It works beautifully for the small, simple islands. The new higher elliptic Gamma function is like a complex, multi-voiced choir. It can sing in many dimensions at once, allowing it to handle the more complicated geometry of these larger number fields. The paper doesn't just guess that these keys exist; it provides a detailed, step-by-step instruction manual on how to assemble them.

The Recipe for Success

The core of the paper is a conjecture—a highly educated guess backed by strong evidence—that if you plug specific numbers into this multi-voiced choir function, the result will be a "unit" (a special kind of number that acts like a building block) inside the number field's extension.

To make this work, the author had to find the perfect "settings" for the experiment. Imagine trying to tune a radio to a clear station; if you are slightly off, you just get static. Similarly, the author defines "optimal settings" where the numbers you plug in are chosen very carefully. In these perfect conditions, the chaotic noise of the calculation disappears, and the result is a clean, algebraic number that fits perfectly into the mathematical structure of the field.

The paper tests this idea by running massive computer simulations. The author writes algorithms to:

Find the right fields: Scanning through thousands of number fields (of degrees 3, 4, 5, and 6) to find the ones that fit the "optimal settings."
Calculate the values: Using the computer to evaluate these complex Gamma functions with extreme precision (up to 1,000 decimal places).
Check the results: Seeing if the calculated numbers match the roots of specific polynomials that define the symmetrical extensions of the fields.

What the Computer Found

The results are promising but not yet a final proof. The paper presents six detailed examples where the math works out beautifully:

Degree 3 (Cubic): The author successfully constructed units for a cubic field, finding that the calculated numbers matched the roots of a polynomial that describes a 4-step extension of the field.
Degree 4 (Quartic): In a quartic example, the method produced a unit that matched a polynomial defining a quadratic extension.
Degrees 5 and 6: The author pushed the method further, successfully computing units for a quintic (degree 5) field and even a degree 6 field. In the degree 6 case, the calculation involved 120 different terms, and the resulting number matched a massive polynomial with coefficients in the billions.

In every single case tested, the "higher elliptic units" calculated by the computer were indeed algebraic numbers that lived in the correct extensions and satisfied a specific formula (the Kronecker limit formula) that links them to the underlying geometry of the field.

The Verdict: A Strong Suggestion, Not a Final Proof

It is important to understand the status of these findings. The paper suggests a powerful new method for solving Hilbert's 12th problem for these specific types of fields, but it does not claim to have proven it mathematically for all cases. The author explicitly states that while the numerical evidence is overwhelming—showing the pattern holds in dozens of complex scenarios—the general proof remains a conjecture.

The paper also rules out the idea that this works for every possible number field immediately. It highlights that the method relies on finding "optimal settings," which are rare and hard to find. For instance, the author notes that for degree 7 fields, they could not find a single example where the conditions were simple enough to run the calculation in a reasonable amount of time.

In short, this paper is a brilliant demonstration that the "multi-voiced choir" of the higher elliptic Gamma function can indeed sing the right notes to unlock these hidden mathematical kingdoms. It provides a concrete, working recipe that has been tested and verified in many difficult cases, offering a hopeful path forward for a problem that has stumped mathematicians for over a century. While the final, universal proof is still waiting to be written, the map provided here is the most detailed and accurate one we have ever seen.

Technical Summary: Computations of Higher Elliptic Units in Optimal Settings

Problem Statement
This paper addresses the construction of generalised elliptic units above number fields $K$ of degree $n \geq 3$ possessing exactly one complex place. The work aims to provide an analytic description of the abelian extensions of such fields, thereby offering a conjectural solution to Hilbert's 12th problem for this specific class of number fields. While the construction of elliptic units for imaginary quadratic fields is well-established via the $\theta$ function and Complex Multiplication, extending this to higher degrees requires the use of higher-dimensional analogues. Specifically, the paper focuses on evaluating "higher elliptic Gamma functions" ( $G_r$ ) at specific algebraic points to generate units in narrow ray class fields.

Methodology
The author employs a computational and conjectural framework based on the following components:

Higher Elliptic Gamma Functions: The construction utilizes the functions $G_r(z, \tau_0, \dots, \tau_r)$ , defined by infinite products and satisfying modular properties related to $SL_{r+2}(\mathbb{Z})$ . These functions generalize the Jacobi $\theta$ function ( $G_0$ ) and the elliptic Gamma function ( $G_1$ ).
Optimal Settings: To make the conjecture tractable and the computations feasible, the paper restricts attention to "optimal settings." These are specific configurations of the number field $K$ $K$ , the modulus $\mathfrak{f}$ $f$ , and the fundamental units $\varepsilon_1, \dots, \varepsilon_{n-2}$ $ε_{1}, \dots, ε_{n - 2}$ that satisfy several restrictive hypotheses:
- The group of totally positive units congruent to 1 modulo $\mathfrak{f}$ coincides with the group of all totally positive units ( $O_{K}^{+, \times} = O_{\mathfrak{f}}^{+, \times}$ ).
- The residue ring $\mathcal{O}_K/\mathfrak{f}$ is cyclic.
- The "content" of the unit systems is 1, and the associated "overflow ideals" are coprime to the smoothing index.
- The number field $K$ contains no proper subfields.
Evaluation Procedure: In these optimal settings, the higher elliptic units are constructed as finite products of ratios of smoothed $G_{n-2}$ $G_{n - 2}$ functions. The evaluation parameters ( $z_\rho, \tau_{j,\rho}$ $z_{ρ}, τ_{j, ρ}$ ) are derived from:
- A compatible set of "base points" (helper ideals).
- The overflow ideals associated with permutations of the fundamental units.
- A smoothing ideal $\mathfrak{a}$ of prime norm $N$ .
  The resulting unit $u_{k,b}$ is defined by a product over the symmetric group $S_{n-2}$ , with the number of terms minimized to $(n-2)!$ .
Computational Techniques:
- Function Evaluation: Since the infinite product definition of $G_r$ is unsuitable for direct computation, the author utilizes a series expansion (Proposition 3.6 in [Nis01]) valid in a "center strip." The algorithm involves a "reorientation step" (using inversion properties to map parameters to the upper half-plane) and a "translation step" (using pseudo-periodicity to map the elliptic variable into the center strip).
- Search Algorithm: A sieving algorithm (Algorithm 1) iterates through polynomials to identify number fields and fundamental units that satisfy the optimal conditions, specifically filtering for small overflow values to ensure computational feasibility.
- Verification: High-precision numerical computations (1000 digits) are performed using PARI/GP. The results are verified by checking if the computed complex numbers are roots of polynomials with coefficients in $K$ (using the LLL algorithm via lindep and algdep) and by confirming the Kronecker limit formula.

Key Contributions

Simplified Conjecture: The paper presents a simplified form of a general conjecture (originally detailed in the author's PhD thesis [Mor26c]) regarding the algebraicity of special values of $G_{n-2}$ functions. This simplified version is tailored specifically to "optimal settings," reducing the complexity of the evaluation parameters and the size of the product.
Explicit Construction: The author provides explicit formulas for the evaluation parameters $z_\rho$ and $\tau_{j,\rho}$ in terms of the fundamental units, overflow ideals, and helper ideals.
Computational Framework: The paper details the specific algorithms required to compute these units, including the handling of the "center strip" for $G_r$ functions and the systematic search for number fields satisfying the restrictive optimal conditions.
Numerical Evidence: The paper provides extensive numerical evidence supporting the conjecture for number fields of degrees 3, 4, 5, and 6.

Results
The paper presents six specific examples where the conjecture is tested:

Degree 3 (Cubic): Two examples are provided. One with a trivial class group ( $q=11$ ) and another with a large class group ( $q=3$ ). In both cases, the computed values are shown to be roots of relative polynomials defining the ray class field, and the Kronecker limit formula is verified.
Degree 4 (Quartic): Two examples are discussed. One with a trivial class group ( $q=2$ ) and another with a non-trivial class group ( $q=7$ ). The computations confirm that the values generate the expected abelian extensions.
Degree 5 (Quintic): An example with $q=3$ is presented, involving a product of 6 smoothed $G_3$ functions. The computation time is noted to vary significantly based on the size of the overflow ideals.
Degree 6: A discussion of a degree 6 example is provided. While the full parameter list is too large to include, the paper confirms that the computed unit matches the root of a high-degree reciprocal polynomial defining the class field.

The computations consistently show that the constructed complex numbers are algebraic units in the expected ray class fields and satisfy the predicted Kronecker limit formula:
$N\zeta'_{\mathfrak{f}}(c, 0) - \zeta'_{\mathfrak{f}}(ac, 0) = \log |u_c|^2$

Significance and Claims
The paper claims to provide a "simplified form" of a conjecture on the construction of generalised elliptic units, making the problem computationally accessible in specific cases. The primary significance lies in:

Hilbert's 12th Problem: Offering a concrete, analytic method (via $G_{n-2}$ functions) to generate abelian extensions for number fields with exactly one complex place, a case where no general solution is currently known.
Extension of Previous Work: Extending the construction of conjectural elliptic units for complex cubic fields (by Bergeron, Charollois, and García [BCG23]) to higher degrees ( $n=4, 5, 6$ ).
Empirical Support: Providing the first substantial numerical evidence for the algebraicity of these higher elliptic units in degrees 4, 5, and 6, supporting the broader conjecture that these values are indeed algebraic units.

The author notes that the "optimal settings" are restrictive and that finding such fields becomes increasingly difficult as the degree increases (e.g., no degree 7 examples satisfying all conditions were found). The paper does not claim to prove the algebraicity of these values but rather to demonstrate their existence and properties through high-precision computation, thereby supporting the underlying conjecture.